Newly published (April 2004) and just what you are looking for:
"Shakespeare's Songbook" by Ross W. Duffin (Prof. of Music at Case Western Reservere University), published by W.W. Norton. The result of 8 years' research; 160 songs included in or mentioned in Shakespeare's plays, with accompanying CD.

All the publisher's details are here: Shakespeare's Songbook

For people in the UK, I know it's available from French's (saw someone enquiring about it and buying one in there only yesterday !)

"What is love ? 'Tis not hereafter,
Present mirth hath present laughter."

To all those who say otherwise, I say "Sneck up !"

(I love that scene, too).

- jeanie